Trigonometry

by Gelfand, I. M.; Saul, Mark

• ISBN13: 9780817639143
• ISBN10: 0817639144
• Format: Paperback
• Copyright: 1999-07-01
• Publisher: SPRINGER - VERLAG

Summary

Trigonometry , a work in the collection of the Gelfand School Program, is the result of a collaboration between two experienced pre-college teachers, one of whom, I.M. Gelfand, is considered to be among our most distinguished living mathematicians. His impact on generations of young people, some now mathematicians of renown, continues to be remarkable. Trigonometry covers all the basics of the subject through beautiful illustrations and examples. The definitions of the trigonometric functions are geometrically motivated. Geometric relationships are rewritten in trigonometric form and extended. The text then makes a transition to the study of algebraic and analytic properties of trigonometric functions, in a way that provides a solid foundation for more advanced mathematical discussions. Throughout, the treatment stimulates the reader to think of mathematics as a unified subject. Like other I.M. Gelfand treasures in the programa?? Algebra , Functions and Graphs , and The Method of Coordinatesa?? Trigonometry is written in an engaging style, and approaches the material in a unique fashion that will motivate students and teachers alike. From a review of Algebra , I.M. Gelfand and A. Shen, ISBN 0-8176-3677-3: "The idea behind teaching is to expect students to learn why things are true, rather than have them memorize ways of solving a few problems, as most of our books have done. [This] same philosophy lies behind the current text by Gel'fand and Shen. There are specific 'practical' problems but there is much more development of the ideas.... [The authors] have shown how to write a serious yet lively book on algebra." a??R. Askey, The American Mathematics Monthly

Table of Contents

Preface	ix
Trigonometry	1	(20)
What is new about trigonometry?	1	(5)
Right triangles	6	(1)
The Pythagorean theorem	7	(2)
Our best friends (among right triangles)	9	(1)
Our next best friends (among right triangles)	10	(2)
Some standard notation	12	(9)
Trigonometric Ratios in a Triangle	21	(20)
Definition of sin α	21	(3)
Find the hidden sine	24	(2)
The cosine ratio	26	(2)
A relation between the sine and the cosine	28	(1)
A bit of notation	28	(1)
Another relation between the sine and the cosine	29	(1)
Our next best friends (and the sine ratio)	30	(2)
What is the value of sin 90°?	32	(1)
An exploration: How large can the sum be?	33	(1)
More exploration: How large can the product be?	34	(1)
More names for ratios	35	(6)
Relations among Trigonometric Ratios	41	(26)
The sine and its relatives	41	(3)
Algebra or geometry?	44	(1)
A remark about names	45	(1)
An identity crisis?	46	(2)
Identities with secant and cosecant	48	(2)
A lemma	50	(1)
Some inequalities	51	(1)
Calculators and tables	52	(1)
Getting the degree measure of an angle from its sine	53	(2)
Solving right triangles	55	(1)
Shadows	56	(1)
Another approach to the sine ratio	57	(10)
Relationships in a Triangle	67	(24)
Geometry of the triangle	67	(1)
The congruence theorems and trigonometry	68	(1)
Sines and altitudes	69	(1)
Obtuse triangles	70	(1)
The Law of Sines	71	(3)
The circumradius	74	(1)
Area of a triangle	75	(3)
Two remarks	78	(1)
Law of cosines	78	(13)
Angles and Rotations	91	(12)
Measuring rotations	91	(2)
Rotation and angles	93	(1)
Trigonometric functions for all angles	94	(4)
Calculations with angles of rotations	98	(3)
Odd and even functions	101	(2)
Radian Measure	103	(20)
Radian measure for angles and rotations	103	(6)
Radian measure and distance	109	(4)
Interlude: How to explain radian measure to your brother and sister	113	(1)
Radian measure and calculators	114	(1)
An important graph	115	(2)
Two small miracles	117	(6)
The Addition formulas	123	(16)
More identities	123	(2)
The addition formulas	125	(1)
Proofs of the addition formulas	126	(1)
A first beautiful proof	127	(3)
A second beautiful proof	130	(9)
Trigonometric Identities	139	(34)
Extending the identities	139	(1)
The Principle of Analytic Continuation: Higher mathematics to the rescue	139	(2)
Back to our identities	141	(2)
A formula for tan (α + β)	143	(2)
Double the angle	145	(3)
Triple the angle	148	(1)
Derivation of the formulas for sin α/2 and cos α/2	149	(2)
Another formula for tan α/2	151	(1)
Products to sums	152	(2)
Sums to products	154	(19)
Graphs of Trigonometric Functions	173	(34)
Graphing the basic sine curve	173	(2)
The period of the function y = sin x	175	(1)
Periods of other sinusoidal curves	176	(2)
The amplitude of a sinusoidal curve	178	(1)
Shifting the sine	179	(3)
Shifting and stretching	182	(1)
Some special shifts: Half-periods	183	(4)
Graphing the tangent and cotangent functions	187	(1)
An important question about sums of sinusoidal functions	188	(1)
Linear combinations of sines and cosines	189	(3)
Linear combinations of sinusoidal curves with the same frequency	192	(2)
Linear combinations of functions with different frequencies	194	(2)
Finding the period of a sum of sinusoidal curves with different periods	196	(1)
A discovery of Monsieur Fourier	197	(10)
Inverse Functions and Trigonometric Equations	207
Functions and Inverse Functions	207
Arcsin: The inverse function to sin	209
Graphing inverse functions	214
Trigonometric equations	217
A more general trigonometric equation	220
More complicated trigonometric equations	222